This question is related to this one. Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything like that). In the representation theory of $G$, one usually considers representations $G\to\mathrm{Aut}_\mathbf{C}(V)$ where $V$ is a Banach space and the map $G\times V\to V$ is continuous, but in practice one also considers (at the very least) continuous representations on Fréchet spaces like $C^\infty(G)$. The question referenced above is essentially about the definition of a $C^\infty$-vector in a continuous representation of $G$ on a Fréchet space. My interpretation of the answers given is that perhaps this is not completely standardized. However, on page 2 of the book Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups by Borel and Wallach, they introduce the notation $C^\infty(M;V)$ for "the space of $C^\infty$ functions of $M$, with values in $V$, endowed with the $C^\infty$-topology." Here $M$ is a smooth manifold which is "countable at infinity" (I'm not sure what that means but I'm sure I'm happy to assume it for my Lie group $G$, if it's not automatic) and $V$ is any Hausdorff quasi-complete locally convex space over $\mathbf{R}$ or $\mathbf{C}$. There is no explanation of this space's definition, or of its topology, which leads me to believe (perhaps mistakenly) that this is something standard. (One would I think care about this for defining the notion of a $C^\infty$-vector in a general continuous representation of $G$ on a locally convex space.)
Is the notion of a $C^\infty$-map from a smooth manifold to a Hausdorff quasi-complete locally convex space over $\mathbf{R}$ or $\mathbf{C}$, and of the (presumably locally convex) topology on the space of such functions, actually standard? If so, is there a convenient reference where this generality is explained?
(I don't think this appears in any Bourbaki volume because I don't think any of them really combine Lie groups with TVS, although if there were such a volume, I don't doubt that it would have the answer to my question.)
Whether or not one cares about such generality in practice is not really my motivation for asking the question (my efforts to thoroughly read Borel-Wallach have been mostly abortive and I honestly don't know whether or not they use this beyond the case of $V$ Fréchet). I'm really curious about this because, in the case where $k$ is a $p$-adic field, $M$ is a paracompact locally $k$-analytic manifold, $V$ is an arbitrary Hausdorff locally convex space over a $p$-adic field, and smooth maps are replaced by locally analytic ones, spaces of functions and their topologies in this generality are both defined and genuinely important in practice (namely in the representation theory of locally $k$-analytic groups on $p$-adic locally convex spaces that aren't just Banach spaces or even Fréchet spaces). But I'm not sure that mimicking the definition in the non-Archimedean case (which uses Banach subspaces of $V$ and the already-defined case of Banach spaces) would really work in the case of smooth manifolds, because somehow defining smooth maps $M\to V$ in extreme generality seems harder to me than defining (locally) analytic ones as there is no issue of second derivatives, etc. (this is perhaps supported by the Wikipedia page on the Gâteaux derivative, which contains the statement "...for Gâteaux differentials...there are several inequivalent ways to formulate their continuous differentiability.")
